Introduction

• Quantum mechanics has been developed as a theory describing microscopic (atomic and subatomic) processes. In textbooks it is sometimes presented as being derivable from classical mechanics by a substitution like

  p ---> -i(h/2p)¶/¶q,

  h = Planck's constant. However, in such a ``derivation'' there always is a certain arbitrariness. It can be justified only by the fact that its result is yielding a useful description of microscopic reality (or, rather, of the measurement results of measurements performed in that domain). For this reason it seems better to introduce quantum mechanics in an axiomatic way, as is done in the following.

Standard formalism of quantum mechanics

• Quantum mechanical state:
  state vector |y> (pure state) or density operator r (mixture).
• Quantum mechanical standard observable (restricting ourselves to discrete non-degenerate spectra):
  Hermitian operator A = S_m a_m E_m, with eigenvalues a_m, and E_m projection operators on its eigenvectors |a_m>, satisfying E_m² = E_m. The set of projection operators {E_m} is called the spectral resolution of A.
• Probability distribution of measurement results of a measurement of standard observable A performed in state |y> or r:

  pm = |<am|y>|2 or pm = Tr rEm.

This is often referred to as the Born rule.
• Time evolution:
  Pure states: Schrödinger equation:

  i(h/2p)¶|y>/¶t = H |y>,

  h = Planck's constant, H = Hamilton operator. In the following we put h = 2p. If H is not explicitly time-dependent, then the solution can be written according to

  |y(t)> = e-iHt |y(0)> .

  Mixtures: Liouville-von Neumann equation:

  i¶r/¶t = [H,r]-, [H,r]- = Hr-rH,

  having for time-independent H the solution

  r(t) = e-iHtr(0) eiHt.

• Remarks:
  
  • This is essentially all there is to the standard formalism of quantum mechanics.
  • Von Neumann's projection postulate
    Sometimes also von Neumann's projection (or reduction) postulate is taken as part of the formalism of quantum mechanics, to the effect that it is assumed that during a measurement of standard observable A the state vector |y> undergoes a discontinuous transition

    |y> ---> |am>

    if the measurement result is a_m.
    I do not consider von Neumann's projection (or reduction) postulate to be either a necessary or a useful property of a quantum mechanical measurement. It is not necessary, because it is a consequence of a certain interpretation of the quantum mechanical state vector (viz. a realist version of the individual-particle interpretation) that is dispensable. It is not useful because by most practical experimental measurement procedures it is not satisfied. For instance, an ideal photon counter detects photons by absorbing them. Hence, ideally the final state of the electromagnetic field is the vacuum state |0> rather than the eigenvector |n> of the photon number observable corresponding to the number n of detected photons. Since photons that have survived the detection process are not registered at all, it follows that the functioning of the photon counter even depends crucially on not satisfying von Neumann's projection postulate: it is operating better to the extent it is violating the projection rule. Consistency problems of quantum mechanical measurement, arising because of von Neumann projection, could better be dealt with by abandoning the postulate as a measurement principle, rather than by ignoring the existence of such well-tried measuring instruments like photon counters.
  • Note, incidentally, that a generalization of von Neumann projection is applicable as a preparation principle, viz. as a procedure describing conditional preparation.
  • Relative frequencies
    Quantum mechanical probabilities p_m are connected with experiment by comparing them with the relative frequencies N_m/N that the value m is obtained when the experiment is performed a large number (N) of times. Thus,

    pm = limN -> ¥Nm/N.

    In order that quantum mechanics be applicable the limit should exist. This is far from self-evident. It requires that the experiment be repeatable so as to make individual preparations identical in a certain sense (consecutive position measurements of an electron freely travelling into outer space will not yield a limit). In general it is taken for granted that the physical conditions for the existence of the limit are satisfied, i.e., that present-day measurements on microscopic systems are within the domain of application of quantum mechanics.
  • (In)compatibility of observables
    The difference between classical and quantum mechanics is embodied by the possibility that, contrary to classical quantities, two quantum mechanical standard observables A and B may be incompatible, their commutator satisfying [A, B]_- ¹ O.
    For compatible observables the definition of a probability distribution can be generalized to a joint probability distribution according to

    pmn = |<amn|y>|2 or pmn = Tr rEmFn,

    where |a_mn> are the joint eigenvectors of the compatible observables, and {E_m} and {F_n} their spectral resolutions. However, it is impossible to find a similar expression that could serve as a joint probability distribution of incompatible standard observables. This implies that within the domain of application of the standard formalism of quantum mechanics a simultaneous or joint measurement of incompatible observables is impossible.

• It is an important recent insight that many quantum mechanical experiments that have actually been performed are outside the domain of applicability of the standard formalism. In particular, the concept of a quantum mechanical observable must be generalized.
• Generalized quantum mechanical observable (restricting ourselves to discrete spectra):
  Resolution of the identity operator I, consisting of a set of positive (better: non-negative) operators M_m, satisfying S_m M_m=I. These operators generate a so-called positive operator-valued measure (POVM). A generalized observable will be indicated as {M_m}.
• Probability distribution of measurement results of generalized observable {M_m} performed in state |y> or r:

  pm = <y|Mm|y> or pm = Tr rMm.

• The generalized formalism of quantum mechanics encompasses the standard one. If all operators M_m of a POVM are mutually commuting projection operators, then the POVM reduces to a projection-valued measure (PVM), and the generalized observable reduces to a standard one (or, in case of degeneracy, to a set of standard ones, viz. all standard observables f(A) having the same spectral resolution).
• Note that for characterizing a generalized observable it is neither necessary nor useful to introduce, analogously to the operator A = S_m a_m E_m of a standard observable, an operator S_m a_m M_m. Actually, both for standard and generalized observables probabilities are determined by m rather than by a_m, the latter's magnitude being irrelevant. This will play an important role in choosing an interpretation of the mathematical formalism of quantum mechanics.
• On the basis of a quantum mechanical description of the interaction of microscopic object and measuring instrument it will be demonstrated that the generalized formalism is a natural extension of the standard one. It has many applications, including the Stern-Gerlach experiment, which, on closer scrutiny, is a paradigm of the standard formalism only in an approximate sense (cf. Publ. 36).
• The generalized formalism will be particularly useful for describing measurements in which information is jointly obtained on incompatible standard observables (deemed impossible by the standard formalism, but nevertheless playing an important role in foundational discussions of quantum mechanics).

• There is a difference between quantum mechanics (the theory) and the reality it is describing (electrons etc.). Electrons are not wave packets flying around in space. Electrons are physical objects, wave packets are theoretical notions. Probably the only way to have a wave packet flying in space is by throwing one's quantum mechanics textbook. If a quantum mechanical observable were a Hermitian operator, one could observe it by looking into one's quantum mechanics textbook. These trivial remarks are nor superfluous because whole generations of physicists have been trained to think about electrons as wave packets flying around in space, and to see values of Hermitian operators as properties of microscopic objects.
• One should be aware of the possibility that quantum mechanics is not the `theory of everything', yielding a complete description of `all there is', but that it may only be yielding a description of certain aspects of microscopic reality, much in the same way as classical mechanics just describes certain aspects of macroscopic reality (maybe an electron is not a wave packet, just like the planet Mars is not the point particle nor the rigid body figuring in textbooks of classical mechanics).
• The domain of application of quantum mechanics is microscopic reality. It is an open question whether this domain can be extended to the macroscopic world. Of course, it is recommendable to explore the applicability of quantum mechanics by applying it to experiments on a wider set of objects than just the microscopic ones (for instance, mesoscopic objects), but there is no warrant that this will keep working all the way up to the macroscopic domain. The idea that quantum mechanics should also be applicable to the macroscopic world is a consequence of a scientific methodology in which theories are supposed to be either true (i.e. universally applicable) or false. However, it is more and more realized that this is not how science works in actual practice: in general a physical theory has a restricted domain of application in which experimental data are described by it in a more or less exact way (in general, less exact as the boundaries of the domain are approached). It is not evident why this should be different for quantum mechanics.
• As quantum mechanics is different from the reality it purports to describe, we need an interpretation. I will use the notion of `interpretation' in the following sense: An interpretation of a physical theory is a mapping from its mathematical formalism into the physical world.
  There are different possibilities for interpreting the quantum mechanical formalism. In the following two of these will play an important role, viz. (cf. figure)
  a: the realist interpretation in which the quantum mechanical formalism is assumed to be mapped into the microscopic reality of the atomic and subatomic world, b: the empiricist interpretation in which the quantum mechanical formalism is thought to be mapped into the macroscopic world of the phenomena corresponding to operations performed on microscopic objects (like flashes on a fluorescent screen, or clicks of a Geiger counter).
  Like the mathematical formalism also its interpretation cannot be derived. An interpretation is neither true nor false. It is either useful or less useful for establishing a correspondence between the mathematics of the theoretical formalism and the physics of the world we are dealing with. For several reasons, to be discussed in the following, the realist interpretation (which is the usual textbook interpretation) turns out to be less useful than the empiricist one.
• In my view quantum mechanics is an ordinary physical theory. I do not think that its domain of application encompasses notions like the human mind, human consciousness, free will, or the `universe as a whole' (although, of course, quantum mechanics certainly is important for cosmology as far as the behaviour of microscopic objects is important to it). Attempts to apply quantum mechanics to such notions will largely be ignored in the following.

• No human observer has ever seen an electron or even an atom. Everything we know about such objects stems from indirect observation by means of measuring instruments, obtaining their information by interacting with the microscopic objects, and amplifying this information to the macroscopic level of the human observer.
• The human observer is as dispensable in quantum mechanics as he (short for `he or she') is in classical mechanics. He sees only the macroscopic parts of his measuring instruments. In present-day practice of the physical science of the microscopic domain human observation is largely restricted to the tables and graphs that have been printed on the basis of data obtained by the scientists computer from a measuring instrument, the measurement results of which having been sent to the computer without any human interference.
• An influence like the reduction (collapse) of the wave packet, allegedly exerted by a human observer on a microscopic object by means of observation, would be equally miraculous as killing a fly by just looking at ones fly swatter.
• In order not to be seduced into any kind of psychophysicalism with respect to observation in quantum mechanics it is advisable to avoid any reference to the human observer; instead it is necessary to carefully consider the interaction between the microscopic object and the measuring instrument mediating the observation.

• It is important to note that in textbooks of quantum mechanics the notion of `measurement' is dealt with in a very insufficient way. In general the measuring instrument is not dealt with as a separate physical object, being different from the microscopic object, and having to interact with that object in order that information be transmitted so as to become observable at the macroscopic level.
• In a quantum mechanical measurement microscopic information is transferred from a microscopic object to a measuring instrument. Such a microscopic process belongs to the domain of quantum mechanics. The part of the measurement process that is to be described by means of quantum mechanics is sometimes referred to as the pre-measurement. It is not clear whether also the amplification process to the macroscopic level is within the domain of application of quantum mechanics. There is no a priori reason to believe it does (unless one thinks that quantum mechanics is the `theory of everything'). The pre-measurement phase is the crucial part of the measurement. For instance, in the Stern-Gerlach experiment the pre-measurement phase is the initial part in which an atom interacts with an inhomogeneous magnetic field, thus establishing a correlation between its spin and its position. Amplification to macroscopic dimensions is realized by putting detectors in the outgoing beams. If these detectors are at positions where the outgoing beams do not overlap anymore, then the amplification may be treated as a classical process.
• Quantum mechanical description of pre-measurement
  In the pre-measurement phase of the measurement of standard observable A (with eigenvalues a_m and eigenvectors |a_m>) the interaction between object and measuring instrument is described by a Schrödinger equation. We restrict ourselves here to a rather simplistic model in which, if the initial state of the object is given by |a_m>, the measurement interaction realizes the transition

  |am> |q0> ---> |ym> |qm>,

  in which |q₀> is the initial state of the measuring instrument and |q_m> are the so-called pointer states, corresponding to the possible post-measurement positions of the pointer of the measuring instrument (indicated by m in figures 1 and 2), and the states |y_m> are normalized states of the microscopic object. Due to the validity of the superposition principle (valid for linear differential equations) the final state |Y_f> of the system object+measuring instrument is given by

  |Yf> = Smcm |ym> |qm>

  if the initial object state was given by |y>= S_mc_m|a_m>.
  The pointer states are generally considered orthogonal:

  <qm|qm'> = dm,m'.

  The states |y_m> depend on the specific interaction of object and measuring instrument, and are not uniquely given. In general they are not orthogonal for different values of m:

  <ym|ym'> ¹ dm,m'.

• The state vector |Y_f> = S_mc_m |y_m> |q_m>, being a superposition of product terms |y_m> |q_m>, is an example of a so-called entangled state. Due to the fact that their origin is of a typically quantum mechanical nature, such states are particularly interesting. In an entangled state correlations between the two subsystems are non-classical.
• Measurements of the first and second kind
  In the literature on the foundations of quantum mechanics attention is often restricted to so-called `measurements of the first kind' for which

  |ym> = |am>.

  Measurements for which |y_m> ¹ |a_m> are sometimes called `measurements of the second kind'. The restriction to measurements of the first kind is an important source of misunderstandings with respect to the nature of quantum mechanical measurement. It is directing the attention into the wrong direction, viz. toward the final state of the microscopic object rather than toward the final state of the measuring instrument (which actually is the observationally relevant entity in case of a measurement).
• The reasons for the above-mentioned restriction to measurements of the first kind are two-fold:
  i) The first reason mainly derives from the Copenhagen interpretation, more particularly its embracement of von Neumann's projection postulate. Unfortunately, this postulate is satisfied by hardly any realistic measurement procedure. Not even the Stern-Gerlach experiment, which is often presented as a paradigm of first kind measurements, is exactly satisfying the projection postulate.
  ii) A second reason derives from the relative simplicity of the assumption of first-kindness, seemingly making it superfluous to determine what is really going on in a quantum mechanical measurement, and allowing to take part in the discussion on the foundations of quantum mechanics without having to invest too much energy.
• Note that in more general treatments of measurement density operators will be necessary to take into account unobserved microscopic fluctuations within the pointer. A more general discussion is given within the context of the generalized formalism.

• Realist interpretation of quantum mechanics
  • Figure 1 In the realist interpretation the interpretational mapping is assumed to be from the mathematical formalism into microscopic reality. The mathematical entities of the theory (state vector |y>, density operator r, standard observable A, and generalized observable {M_m}) are assumed to represent properties of the microscopic object. Thus, the quantum mechanical momentum operator is assumed to describe the physical momentum of an electron much in the same way as the classical quantity mv is assumed to describe the physical momentum of a billiard ball. Instruments of preparation, used to prepare the microscopic object in an initial state, as well as measuring instruments, are not represented in the quantum mechanical description by the theory, even if they are physically present. This is symbolized in figure 1 by dashing the preparing and measuring apparata.
  • We should distinguish objectivistic and contextualistic versions of a realist interpretation of quantum mechanics. In an objectivistic-realist interpretation the quantum mechanical description is assumed to refer to objective reality, that is, a reality independent of any `observer including his measuring instruments'. In a contextualistic-realist interpretation quantum mechanical concepts are assumed to have a meaning only within a certain physical context (like the object's environment, or a measurement arrangement).
  • Contrary to a rather common belief, classical theories do not allow an objectivistic-realist interpretation in general. Due to its atomic constitution, and the possible vibrations of its atoms, a billiard ball is not a rigid body at all. At most, a billiard ball is satisfying classical rigid body theory only within contexts (e.g. of experiments) allowing the ball to `behave as if it were a rigid body'. That is, as far as classical rigid body theory can be interpreted realistically, it allows at most a contextualistic-realist interpretation. If the ball is hit so hard that the atoms are set in violent vibrational motion, then classical rigid body theory is not applicable any more.
• Empiricist interpretation of quantum mechanics
  • Figure 2 In the empiricist interpretation the interpretational mapping of |y>, r, A, and {M_m} is assumed to be from the mathematical formalism into the macroscopic reality of sources for preparing microscopic objects and instruments for performing measurements ('phenomena' are assumed to correspond, e.g., to knob settings of preparing instruments, and to pointer positions of measuring instruments). Thus, wave functions and density operators are assumed to be symbolic representations (labels) of preparation procedures (for instance, referring to a cyclotron with specified knob settings as a preparing instrument of a beam of particles; but they may also refer to less specified natural processes like the preparation of an electromagnetic field by the sun); a quantum mechanical observable is assumed to be a label of a measuring instrument or measurement procedure (for instance, a photosensitive device for detecting photons). Even though the microscopic object is present, in the empiricist interpretation it is assumed not to be represented by the quantum mechanical formalism. This is symbolized in figure 2 by dashing the object. In the empiricist interpretation the quantum mechanical formalism is assumed to describe `just the phenomena', `phenomena' being located within the macroscopic sources for preparing microscopic objects and within the instruments for measuring their properties.
  • In a sense the empiricist interpretation is a realist one, too, since it is a mapping of the mathematical formalism into physical reality, be it into the macroscopic (directly observable) part of this reality rather than the microscopic one.
  • Notwithstanding similarities should the empiricist interpretation of quantum mechanics be carefully distinguished from the empiricist views as fostered by the philosophical doctrine of logical positivism, considering metaphysical anything that is unobservable. In particular, an empiricist interpretation of quantum mechanics is perfectly consistent with a belief in the "real" existence of microscopic objects like atoms and electrons (`reality behind the phenomena'), even though these are not directly observed. The relevant point is that, according to the empiricist interpretation, quantum mechanics does not describe these objects, but it merely describes the relations between the preparing and measuring procedures mediated by them. Far from embracing an anti-realist philosophy denying any `reality behind the phenomena', the empiricist interpretation of quantum mechanics accepts the possibility of such a reality analogously to the way it accepts the reality of atoms within a billiard ball.
  • In order to describe the microscopic objects themselves new (subquantum) theories have to be developed, quite analogously to the necessity of developing atomic solid state theories as subtheories for classical rigid body theory. It should be noted that by allowing the possibility of subquantum ("hidden-variables") theories the empiricist interpretation of quantum mechanics leaves considerably more room for subquantum theories than is left by the realist interpretation of quantum mechanics. This holds true as well for classical rigid body theory, which, if interpreted in a realist sense, would yield a rather absurd model of a billiard ball as a closely packed configuration of rigid atoms.
  • The empiricist interpretation should also be distinguished from the Copenhagen interpretation, which, although having an empiricist reputation, actually contains many realist elements stemming from an inclination to view quantum mechanics as a description of microscopic reality itself (for instance, von Neumann's projection postulate). Nevertheless, the empiricist interpretation is indebted to the Copenhagen one by taking seriously the importance attributed to the role of the measuring instrument in assessing the meaning of the quantum mechanical formalism, be it without neglecting the distinction between the realities of microscopic object and measuring instrument while being engaged with quantum mechanical measurement.
• Reasons to prefer a realist interpretation of quantum mechanics
  • Ontological reason:
    • Quantum reality
      According to our observations there seems to be a fundamental difference between macroscopic and microscopic reality, the latter being experienced as ``strange and paradoxical''. Such ``strangeness'' is incorporated into the quantum mechanical formalism (entanglement, incompatibility), which, for this reason, might be thought to yield a description of a microscopic quantum reality.
      Such strangeness of microscopic reality has an appeal of its own, lacking in the empiricist interpretation. It is widely felt that such a strange quantum reality may open up new vistas for understanding, and for future applications of quantum mechanics (quantum computing, quantum information).
  • Epistemological reason:
    • The classical paradigm
      Classical mechanics is generally thought to be successfully interpretable in a realist sense (however, compare). From this point of view it is not unreasonable to try to analogously interpret quantum mechanics in a realist way (just changing the notions of state, physical quantity, and the equation governing time evolution, but maintaining as much as possible the classical way of thinking). Hence, the classical paradigm favours a realist interpretation of quantum mechanics.
    • Platonism
      In Plato's allegory of the cave it is realized that our knowledge about reality may be compared with the knowledge of cave dwellers who are able to look only into the inward direction of the cave, and who obtain knowledge about what is happening outside the cave only by means of the shadows cast by the objects of the outside world on the cave's inside wall. The upshot is that what we see is just an imperfect image of reality. The difference between a realist and an empiricist interpretation of quantum mechanics can be characterized in terms of this allegory by asking whether the theory is describing either the real (outside) quantum world, or just its shadows, the "phenomena". Plato would consider unscientific any theory aspiring to just describe the shadows. It would be in agreement with Plato to assume that quantum mechanics could only derive its scientific status from the deep insights it gives with respect to the constitution of microscopic reality. Platonic idealism favours a realist interpretation of physical theories.
  • Methodological reason:
    • Inference to the best explanation
      The hypothesis that `reality is like quantum mechanics says it is' is often assumed to provide the best explanation of the success of quantum mechanics. Here `reality' means the `reality of the microscopic object itself' (rather than `just the phenomena'). It is felt that an empiricist interpretation is renouncing too much the possibilities of quantum mechanics as an explanatory device. For instance, atoms are stable objects because they satisfy quantum mechanics (rather than classical theory); it would be absurd to assume that atoms are stable as a consequence of measurements which are performed on them (John Bell: ``To restrict quantum mechanics to be exclusively about piddling laboratory operations is to betray the great enterprise.''). Also Einstein criticized the Copenhagen interpretation for resigning to contextualism, and for abandoning the requirement that a physical theory describe objective reality rather than a reality that is interacting with a measuring instrument.
• Reasons to prefer an empiricist interpretation of quantum mechanics
  Ontological reason:
  • We just do not observe microscopic objects but pointer positions of measuring instruments
    The main reason to prefer an empiricist interpretation of quantum mechanics is that any experimental test of quantum mechanics compares the probability distributions of quantum mechanics with experimental data (relative frequencies) obtained by letting a microscopic object interact with a measuring instrument and subsequently observing a phenomenon corresponding to the state the measuring instrument is in (so-called `pointer readings'). We expect the pointer position to tell us something about the microscopic object, but we should be aware that a pointer reading of a macroscopic measuring instrument is not a direct determination of a property of the microscopic object.
    Distinguishing between `pointer readings' and `properties of the microscopic object' solves one of the ontological riddles of quantum mechanics, namely that it would be impossible to attribute the measurement result to the microscopic object as a property possessed already prior to measurement. For instance, in a position measurement it would be impossible to infer from a click of the particle detector that this is a consequence of the particle being there before the measurement event (Jordan: ``By measuring we force the electron to take up a particular position; previously it was neither here nor there; it had not yet decided about taking up any particular position.''). Within the realist interpretation this is corroborated by the inapplicability of the `possessed values' principle . Nevertheless, the idea that microscopic objects do not have properties (like position) unless such a property is measured, is rather counterintuitive.
    In the empiricist interpretation the problem does not arise, because it is completely natural that the measurement result (the post-measurement pointer position) comes into being only during the measurement, and, hence, cannot be attributed to the microscopic object as a property possessed prior to measurement. According to the empiricist interpretation such properties are not even described by quantum mechanics; they require a subquantum theory for their description. Hence, from this perspective there is no reason to doubt that an electron had a well-defined position previous to measurement, be it that quantum mechanics does not tell us all about it (and analogously for the other observables).
  • Epistemological reason:
    Analogous cases
    Consider e.g. classical rigid body theory, which, as argued here, does not allow an objectivistic-realist interpretation, although a contextualistic-realist one might be feasible. However, on closer scrutiny it turns out that classical rigid body theory should better be interpreted in an empiricist than in a contextualistic-realist way. Indeed, due to its atomic constitution a billiard ball is not really a rigid body, even if no deviation from rigidity is observed. Classical rigid body theory just describes the `phenomena within its domain of application' even if there are atomic vibrations, provided these are imperceptibly small. Within the domain of application of classical rigid body theory the atomic constitution may become important if e.g. optical measurements are performed.
    Another analogy is provided by thermodynamics, which theory describes thermal phenomena within its domain of application (determined by the requirement that a condition of `molecular chaos' be satisfied, which here defines the context). Here, too, the common usage of considering `temperature' as a property of an object (rather than as a pointer reading on a thermometer scale) is questionable (thus, if temperature were objectively defined by kinetic energy, a way to get ones tea water boiling would be to drop ones teapot from a sufficient height). Evidently, even a contextualistic-realist interpretation of thermodynamics has its problems, which can be solved by an empiricist one (doubtless, a thermometer dropped together with the object will not show the increase of temperature mentioned in the above example).
    In the following it will be seen that similar problems exist for the interpretation of quantum mechanics: also here a contextualistic-realist interpretation, as involved in the Copenhagen ideas of correspondence and complementarity, is not sufficient to solve all interpretational problems; an empiricist interpretation is better suited for that purpose.
  • Methodological reasons:
    • Evading the paradoxes of a realist interpretation
      The failure of the `possessed values' principle is one of the many problems and paradoxes that plague the realist interpretation (for instance, particle-wave duality, the "measurement problem", EPR nonlocality). Evading these paradoxes is an important reason to prefer the empiricist interpretation, in which they do not arise.
      Realist interpretations are inexhaustible sources of speculative ideas about microscopic reality. It is questionable whether all these ideas, although very exciting, are equally fruitful. Some of these are rather reminiscent of ideas about the world aether, made obsolete by the (empiricist!) development of relativity theory (thus, EPR nonlocality is as nonempirical as 100 years ago was the world aether). Sticking to an empiricist interpretation of quantum mechanics is a way to evade being distracted from our experimentally relevant observations of pointer positions toward an imagined world of properties attributed to microscopic reality on the basis of the peculiarities of the quantum mechanical formalism.
    
    
    • Escaping from suggestions implied by the standard formalism
      The applicability of the generalized formalism to quantum mechanical measurements is an important indication of the usefulness of an empiricist interpretation. As a matter of fact, experimental data are never crucially dependent on the (eigen)values of observables because they refer to pointer positions of a measuring instrument, the labeling of which is man-made, and, for this reason, quite arbitrary (this is symbolized by denoting labels by m rather than by a_m). Generalized observables (represented by POVMs) are independent of the precise way their values are labelled. Both for standard and for generalized observables the probability distributions are only dependent on m, not on a_m (there is no operator S_m a_m M_m to provide eigenvalues).
      For relations that do depend on the (eigen)values (like the Heisenberg-Kennard-Robertson inequalities or the Bell inequalities) there exist alternatives which are independent of them, and which yet have essentially the same physical meaning (e.g. the entropic uncertainty relations and the BCHS inequalities, respectively).
    
    
    • Acknowledging the essential role of measurement
      In a realist interpretation of quantum mechanics there is a tendency to ignore the role played by measurement in assessing the meaning of the quantum mechanical formalism. This role was stressed by the founding fathers of quantum mechanics (compare the Copenhagen interpretation), however without meeting much response in textbooks of quantum mechanics, in which the theory is generally presented in an objectivistic-realist way in which the measuring instrument is completely left out of consideration. In an empiricist interpretation the role of measurement cannot be ignored. By exploiting the generalized formalism of quantum mechanics it can be seen that the Copenhagen emphasis on `measurement' was not without a physical basis. It, presumably, is not very wise to ignore the role of measurement (as well as preparation) in accounting for the strangeness of quantum mechanics. By attributing properties of the quantum mechanical formalism to the reality behind the phenomena rather than to preparations and measurements, we take the risk to fool ourselves by mistaking in Plato's allegory of the cave the shadows for reality. Contrary to Plato's contention we probably are able to deal in a scientific way only with the shadows.
  
  
  
  Realist versus instrumentalist interpretation of quantum mechanics
  • In the literature on the philosophy of quantum mechanics the choice is often between a realist or an instrumentalist interpretation rather than between a realist or an empiricist one.
  • In an instrumentalist interpretation the mathematical formalism of quantum mechanics is considered as "just an instrument" for calculating measurement results. No physical meaning is attributed to either state vector or observable.
  • In the past instrumentalism has been an acceptable way to circumvent the problems and paradoxes raised by a realist interpretation. However, instrumentalism suffers from too large a vagueness. By not specifying what is meant by `measurement results' different possibilities are left open. In particular, not specifying whether an observable is interpreted in a realist or an empiricist way has caused quite a lot of confusion (compare the difference between EPR and Bell experiments).
  • By not attributing quantum mechanical concepts as properties to the microscopic object an empiricist interpretation of quantum mechanics is gaining the same advantage as an instrumentalist one in dealing with the problems and paradoxes of quantum mechanics. This is achieved without introducing any of the confusing vagueness characterizing the instrumentalist interpretation. Therefore in my opinion the instrumentalist interpretation of quantum mechanics is obsolete by now.
  • At least in Bohr's view, the Copenhagen interpretation is instrumentalist with respect to the state vector. Unfortunately, in many textbooks of quantum mechanics the way the state vector is dealt with can hardly be distinguished from a realist one, even when adherence to the Copenhagen (orthodox) interpretation is acknowledged. A strictly instrumentalist version of the Copenhagen interpretation would not be liable to the paradoxes stemming from this inclination towards realism. In particular, von Neumann's projection postulate would not have the undesirable features it may have in a realist interpretation (like EPR nonlocality). However, a more promising way is possible, viz. by upgrading the Copenhagen interpretation to an empiricist interpretation liable to be called a neo-Copenhagen interpretation (cf. Publ. 52).
  
  
  
  Ontic versus epistemic interpretation of quantum mechanics
  • Sometimes a distinction is drawn between an ontic and an epistemic interpretation of quantum mechanics, the latter meaning that not reality itself is described (as is understood in the ontic interpretation), but that the theory is yielding `just a description of our knowledge of that reality'.
  • This terminology is used in order to stress that quantum mechanics is a physical theory rather than a psychological one, the latter allegedly being implied by the epistemological appearance of an ensemble interpretation of the quantum mechanical state vector. By doing so, it is suggested that an ensemble interpretation of quantum mechanics is not really physics.
  • The dichotomy of `ontic versus epistemic interpretations' can be seen as an attempt to remedy the vagueness of the `realist/instrumentalist dichotomy' by stressing the `ontic character of reality' as well as the `epistemic character of the reference of the instrumentalist interpretation to measurement results' (often equated with `sensations stored in our minds'). However, as seen from the possibility of an empiricist interpretation, do measurement results also have an ontic character (viz. as pointer positions of measuring instruments).
  • In my view the dichotomy `ontic versus epistemic' is potentially misleading. Independently of its interpretation, any physical theory is a representation of our knowledge about a certain physical domain, and, hence, epistemic. On the other hand, as far as an interpretation of quantum mechanics is a mapping of its mathematical formalism into reality, any interpretation of that theory is ontic (in the sense that the theory is thought to describe some part of physical reality). In particular, the `ontic versus epistemic' dichotomy does not distinguish between the realist and empiricist interpretations, which are both ontic in the above sense (even though mapping into different parts of reality).
  • In order to avoid misunderstanding I shall avoid this dichotomy. Instead, in the following a distinction is drawn between `individual-particle' and `ensemble' interpretations of the quantum mechanical state vector, implying that in both interpretations the interpretational mapping is from the theory into reality, be it that in the ensemble interpretation the reality is not that of a single (individual) microscopic object, but just the reality as encountered in an ensemble of individual measurements (defining the quantum mechanical probabilities p_m as relative frequencies in the ensemble).
  • Far from being `just psychological', ensembles are dealt with in experimental quantum mechanical practice by repeating a measurement of a property of an individual object a large number of times under conditions warranting the existence of a limit to the relative frequencies N<sub>m</sub>/N. Reservations based on the `possessed values' principle as regards the theoretical feasibility of ensembles apply only to the objectivistic-realist interpretation in which quantum mechanical measurement results are considered to be objective properties of microscopic objects. Such reservations are futile in the contextualistic-realist interpretation as well as in the empiricist interpretation of quantum mechanics, where the `possessed values principle' is assumed not to be valid.